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arXiv math-ph/0201041math-phmath.MPmath.SP
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随机爆破分形格上的薛定谔算子

Schrödinger operators on fractal lattices with random blow-ups

Christophe Sabot

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15 pages
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摘要：

从一个离散的自相似集$X$出发，通过对初始集$X$进行爆破，我们可以构造一个无界集$X_{<\infty>}$。我们考虑随机爆破和证明的频谱的自然拉普拉斯算子$X_{<\infty>}$（和相关的晶格）的基本性质。我们证明了该算子的谱型几乎必然决定于爆破，并且该谱几乎必然与态密度的支集相一致（实际上我们的结果更精确）.我们还证明了，如果态密度完全由所谓的Neuman-Dirichlet本征值产生，那么谱几乎必然是具有紧支本征函数的纯点.

英文摘要：

Starting from a finitely ramified self-similar set $X$ we can construct an unbounded set $X_{<\infty>}$ by blowing-up the initial set $X$. We consider random blow-ups and prove elementary properties of the spectrum of the natural Laplace operator on $X_{<\infty>}$ (and on the associated lattice). We prove that the spectral type of the operator is almost surely deterministic with the blow-up and that the spectrum coincides with the support of the density of states almost surely (actually our result is more precise). We also prove that if the density of states is completely created by the so-called Neuman-Dirichlet eigenvalues, then almost surely the spectrum is pure point with compactly supported eigenfunctions.

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